Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem

Ahn, Kook Jin; Guha, Sudipto

doi:10.1007/978-3-642-22012-8_42

Cited by 41 publications

(71 citation statements)

References 29 publications

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“…Comparison to Other Results: Efficient approximation schemes for bipartite graphs were considered in [2] -that paper also provided inefficient (Ω(n 1/δ m) time) algorithms for the non-bipartite case. In a companion paper [3] we consider uncapacitated bmatching problems, specifically the weighted bipartite case and the unweighted non-bipartite case.…”

Section: Width and Approximationsmentioning

confidence: 99%

“…We now have a vertex set V (2) . Set y implies a b-matching M (2) in G (2) of same weight (merge edges). (ii) Matching M (2) implies a bmatching M (1) in G (1) of same weight (merge vertices).…”

Section: Rounding Uncapacitated B-matchingsmentioning

confidence: 99%

“…(1) ij } satisfies LP1(b (1) ) over V , then {y (2) ij } satisfies LP1(b (2) ) over G (2) and i,j w ij y (2) …”

Section: Lemma 72 (Second Phase) If {Ymentioning

confidence: 99%

“…Mestre [26] provided a ( [26]. For the bipartite case, (1 − δ) approximation schemes were provided in [2] (see the end of the section for more detailed comparison). It is known that solving the bipartite relaxation for the weighted b-Matching problem within a (1 − δ) approximation (for any δ > 0) will always produce a ( 2 3 −δ)-approximation algorithm for general nonbipartite graphs [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…After scaling we have (1 − δ) j y (1) ij ≤ j y (1) ij − δt ≤ j y (1) ij − 2 ≤ b (2) i − 1 from the definition of b (2) in line (3b) of Algorithm 4. Therefore the new vertices cannot be in any violated vertex or set constraint; from the first part of Lemma 7.1 (now applied to LP1(b (1) ) instead of LP1(b)).…”

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confidence: 99%

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Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b–Matching Problems in Nonbipartite Graphs

Ahn

Guha

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present the first fully polynomial approximation schemes for the maximum weighted (uncapacitated or capacitated) b-Matching problem for nonbipartite graphs that run in time (near) linear in the number of edges, that is, given any δ > 0 the algorithm produces a (1−δ) approximation in O(m poly(δ −1 , log n)) time. We provide fractional solutions for the standard linear programming formulations for these problems and subsequently also provide fully polynomial (near) linear time approximation schemes for rounding the fractional solutions.Through these problems as a vehicle, we also present several ideas in the context of solving linear programs approximately using fast primal-dual algorithms. First, we show that approximation algorithms can be used to reduce the width of the formulation, and as a consequence we induce faster convergence. Second, even though the dual of these problems have exponentially many variables and an efficient exact computation of dual weights is infeasible, we can efficiently compute and use a sparse approximation of the dual weights using a combination of (i) adding perturbation to the constraints of the polytope and (ii) amplification followed by thresholding of the dual weights.These algorithms also have the advantage that they use O(n poly(δ −1 , log n)) storage space and only make O(δ −4 log 2 (1/δ) log n) (or better) passes over a read only list of edges. These algorithms therefore can be run in the semi-streaming model and serve as exemplars where algorithms and ideas developed for the streaming model gives us algorithms for combinatorial optimization problems that were not known in absence of the streaming constraints. *

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Section: Width and Approximationsmentioning

confidence: 99%

Section: Rounding Uncapacitated B-matchingsmentioning

confidence: 99%

“…(1) ij } satisfies LP1(b (1) ) over V , then {y (2) ij } satisfies LP1(b (2) ) over G (2) and i,j w ij y (2) …”

Section: Lemma 72 (Second Phase) If {Ymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b–Matching Problems in Nonbipartite Graphs

Ahn

Guha

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Submodular Maximization Meets Streaming: Matchings, Matroids, and More

Chakrabarti

Kale

2014

Integer Programming and Combinatorial Optimization

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We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching (MCM). We give two incomparable algorithms for this problem with space usage falling in the semistreaming range-they store only O(n) edges, using O(n log n) working memory-that achieve approximation ratios of 7.75 in a single pass and (3 + ε) in O(ε −3 ) passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM.In the sequel, we give generalizations of these results where the maximization is over "independent sets" in a very general sense. This generalization captures hypermatchings in hypergraphs as well as independence in the intersection of multiple matroids.

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